Over the real numbers with /2-coefficients, we compute the C₂-equivariant Borel motivic cohomology ring, the Bredon motivic cohomology groups and prove that the Bredon motivic cohomology ring of the real numbers is a proper subring in the RO(C₂× C₂)-graded Bredon cohomology ring of a point. This generalizes Voevodsky's computation of the motivic cohomology ring of the real numbers to the C₂-equivariant setting. These computations are extended afterwards to any real closed field.
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Deng et al. (2024) studied this question.
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